Questions tagged [locally-ringed-spaces]
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23 questions
17
votes
2
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Functor of points vs. locally/spectrally ringed spaces/$\infty$-topoi
In the following, whenever I say category I mostly really mean $(\infty,1)$-category;
There is essentially two ways of studying (Derived) Algebraic Geometry:
The functor of points viewpoint: One ...
- 367
2
votes
0
answers
136
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Canonicity in split sequence in cotangent spaces
Let $X$ be a locally ringed space. We have for a point $p$ the exact sequence
$$0\to \mathfrak{m}_p^2\to \mathfrak{m}_p\to \mathfrak{m}_p/\mathfrak{m}_p^2 \to 0$$
where $\mathfrak{m}_p$ is the maximal ...
- 167
9
votes
2
answers
473
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Cartesian-closed full subcategory of locally ringed spaces containing smooth manifolds
This coming fall, I will be teaching a course on differential topology to a small group of strong students. In preparation for it, I'm trying to find a category $\mathrm{GDiff}$ with the following ...
- 8,050
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0
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Is the category of diffeological spaces a full subcategory of locally ringed spaces?
It is known that the natural functor of smooth functions from the category of smooth manifolds into the category of locally ringed spaces is a full embedding (see, for example, here).
Is a similar ...
- 8,050
5
votes
3
answers
769
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Sheaf of chain complexes glued by chain homotopy equivalences
Let $(X,\mathcal O_X)$ be a locally ringed space with an open covering $\mathscr U$. Suppose:
For any $U\in\mathscr U$, we have a chain complex $(C_U, d_U)$ such that $C_U$ is an $\mathcal O_X(U)$-...
- 2,709
3
votes
0
answers
164
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References on topological ringed spaces
This is a follow up to this question of mine.
First of all, let me fix some terminologies, which may or may not be standard:
Definition: A topological ringed space is a pair $X := (|X|, \mathcal{O}_X)...
- 1,728
7
votes
0
answers
292
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Does a field extension define an effective descent morphism for locally ringed spaces?
Let $K'/K$ be an extension of fields and set $X=\operatorname{Spec}(K)$ and $X'=\operatorname{Spec}(K')$. As the category of locally ringed spaces has fibre products (see arXiv:1103.2139 or here) we ...
- 71
6
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1
answer
372
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Hakim's definition of a locally ringed topos
In Hakim's book "Topos annelés et schémas relatifs", Chap. III, Def. 2.3 states that a ringed topos $(X,A)$ is a locally ringed topos when two equivalent conditions are satisfied:
(i) For ...
- 65.2k
2
votes
0
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83
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For which locally ringed spaces is the structure sheaf given by LRS morphisms to the real line?
Let $\mathsf{LRS}_{\mathbb R}$ denote the category of locally $\mathbb R$-ringed spaces.
Given a locally ringed space $(X,\mathcal O_X)$, write $C_{(X,\mathcal O_X)}^p$ for the hom-sheaf on $X$ of ...
- 10.7k
2
votes
3
answers
447
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Localification of a ringed space
Call a ringed space local it if it lies in the image of the obvious faithful, non-full functor from locally ringed spaces to ringed spaces.
Given a ringed space, is there a map $f$ from it to some ...
user141346
5
votes
3
answers
579
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Locally ringed space with noetherian stalks and a non-coherent structural sheaf
I am looking for a locally ringed space the stalks of which are noetherian and such that the structural sheaf is not coherent over itself. Can you provide me an example of this?
Notice that one may ...
- 549
3
votes
0
answers
419
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Immersions of locally ringed spaces and locally closed image
Let $f:X\to Y$ be a morphism of locally ringed spaces. In this MSE answer, the first definition below is suggested.
Say $f:X\to Y$ is an $R$-immersion of locally ringed spaces if it's a topological ...
- 10.7k
6
votes
1
answer
890
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Morphisms of locally ringed spaces into affine schemes
In Görtz and Wedhorn's Algebraic Geometry I, there's the following proposition:
Proposition 3.4. Let $(X,\mathcal O_X)$ be a locally ringed space. If $Y$ is an affine scheme then the natural map ...
- 10.7k
11
votes
1
answer
1k
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About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$
I've asked this question https://math.stackexchange.com/questions/1407451/about-the-relation-between-the-categories-textsch-textlrs-and-text on math.stackexchange , however I don't think I will ...
- 2,275
1
vote
0
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244
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Embedding dimension: local finiteness & intuition for more general spaces
Can every complex analytic space be covered by Stein spaces of finite embedding dimension?
I am almost sure that ought to be true. Here the definition of embedding dimension I have in mind is
$$
\...
- 2,031
14
votes
1
answer
2k
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Which local ringed spaces are schemes?
(This was originally asked on math.stackexchange, but didn't get any responses. I figured it might be worthwhile to move it here and try again.)
This paper gives a proof that the underlying ...
- 659
4
votes
3
answers
699
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Are schemes pushouts of neighbourhoods and formal neighbourhoods?
Hello,
I have two questions, the first less important.
Let $X$ be a scheme, $x \in X$ a schematic point.
What is an elegant way of defining/characterizing the map $\operatorname{Spec}(O_{X,x}) \to ...
- 5,632
23
votes
0
answers
2k
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Riemannian manifolds etc. as locally ringed spaces?
There are, among others, three general ways of equipping a "space" (which for the purposes of this question could be a topological space or a differentiable manifold, according to the case) with ...
- 23.7k
12
votes
3
answers
4k
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Justification of the term "invertible sheaf"
Let $X$ be a locally ringed space (or a scheme) and $M,N$ two $\mathcal{O}_X$-modules such that $M \otimes N \cong \mathcal{O}_X$. Does it follow that $M$ is invertible in the usual sense, namely that ...
- 65.2k
11
votes
1
answer
1k
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Examples of locally ringed spaces
I want to know more classes of examples of locally ringed spaces. The reason is that when I want to prove/disprove something about locally ringed spaces, my examples are often not eclectic enough. ...
- 65.2k
8
votes
2
answers
4k
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Closed subschemes and pulling back the structure sheaf via the inclusion map
I would just like a clarification related to closed subschemes.
If $(X,{\cal O}_X)$ is a locally ringed space and $A\subset X$ is any subset with the subspace topology then $i^{-1}{\cal O}_X$ will be ...
- 2,844
6
votes
2
answers
995
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Given a morphism from X to Y, when is the morphism from O_Y to the pushforward of O_X injective
I would like to know under what condition the morphism $\mathcal{O}_Y\longrightarrow f_\ast \mathcal{O}_X$ induced by a morphism $f:X\longrightarrow Y$ of schemes is injective.
Let me give an example ...
- 9,697
36
votes
3
answers
4k
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What is the right version of "partitions of unity implies vanishing sheaf cohomology"
There are several theorems I know of the form "Let $X$ be a locally ringed space obeying some condition like existence of partitions of unity. Let $E$ be a sheaf of $\mathcal{O}_X$ modules obeying ...
- 161k